insertIdx

Proves various lemmas about List.insertIdx.

@[simp]

theorem List.insertIdx_zero {α : Type u} {xs : List α} {x : α} : xs.insertIdx 0 x = x :: xs

@[simp]

theorem List.insertIdx_succ_nil {α : Type u} {n : Nat} {a : α} : [].insertIdx (n + 1) a = []

@[simp]

theorem List.insertIdx_succ_cons {α : Type u} {xs : List α} {hd x : α} {i : Nat} : (hd :: xs).insertIdx (i + 1) x = hd :: xs.insertIdx i x

theorem List.length_insertIdx {α : Type u} {a : α} {i : Nat} {as : List α} : (as.insertIdx i a).length = if i ≤ as.length then as.length + 1 else as.length

theorem List.length_insertIdx_of_le_length {α : Type u} {i : Nat} {as : List α} (h : i ≤ as.length) (a : α) : (as.insertIdx i a).length = as.length + 1

theorem List.length_insertIdx_of_length_lt {α : Type u} {as : List α} {i : Nat} (h : as.length < i) (a : α) : (as.insertIdx i a).length = as.length

@[simp]

theorem List.eraseIdx_insertIdx {α : Type u} {i : Nat} {l : List α} (a : α) : (l.insertIdx i a).eraseIdx i = l

theorem List.insertIdx_eraseIdx_of_ge {α : Type u} {a : α} {i m : Nat} {as : List α} : i < as.length → i ≤ m → (as.eraseIdx i).insertIdx m a = (as.insertIdx (m + 1) a).eraseIdx i

theorem List.insertIdx_eraseIdx_of_le {α : Type u} {a : α} {i j : Nat} {as : List α} : i < as.length → j ≤ i → (as.eraseIdx i).insertIdx j a = (as.insertIdx j a).eraseIdx (i + 1)

theorem List.insertIdx_comm {α : Type u} (a b : α) {i j : Nat} {l : List α} : i ≤ j → j ≤ l.length → (l.insertIdx i a).insertIdx (j + 1) b = (l.insertIdx j b).insertIdx i a

theorem List.mem_insertIdx {α : Type u} {a b : α} {i : Nat} {l : List α} : i ≤ l.length → (a ∈ l.insertIdx i b ↔ a = b ∨ a ∈ l)

theorem List.insertIdx_of_length_lt {α : Type u} {l : List α} {x : α} {i : Nat} (h : l.length < i) : l.insertIdx i x = l

@[simp]

theorem List.insertIdx_length_self {α : Type u} {l : List α} {x : α} : l.insertIdx l.length x = l ++ [x]

theorem List.length_le_length_insertIdx {α : Type u} {l : List α} {x : α} {i : Nat} : l.length ≤ (l.insertIdx i x).length

theorem List.length_insertIdx_le_succ {α : Type u} {l : List α} {x : α} {i : Nat} : (l.insertIdx i x).length ≤ l.length + 1

theorem List.getElem_insertIdx_of_lt {α : Type u} {l : List α} {x : α} {i j : Nat} (hn : j < i) (hk : j < (l.insertIdx i x).length) : (l.insertIdx i x)[j] = l[j]

@[simp]

theorem List.getElem_insertIdx_self {α : Type u} {l : List α} {x : α} {i : Nat} (hi : i < (l.insertIdx i x).length) : (l.insertIdx i x)[i] = x

theorem List.getElem_insertIdx_of_gt {α : Type u} {l : List α} {x : α} {i j : Nat} (hn : i < j) (hk : j < (l.insertIdx i x).length) : (l.insertIdx i x)[j] = l[j - 1]

@[reducible, inline, deprecated List.getElem_insertIdx_of_gt (since := "2025-02-04")]

abbrev List.getElem_insertIdx_of_ge {α : Type u_1} {l : List α} {x : α} {i j : Nat} (hn : i < j) (hk : j < (l.insertIdx i x).length) : (l.insertIdx i x)[j] = l[j - 1]

Equations

• @List.getElem_insertIdx_of_ge = @List.getElem_insertIdx_of_gt

theorem List.getElem_insertIdx {α : Type u} {l : List α} {x : α} {i j : Nat} (h : j < (l.insertIdx i x).length) : (l.insertIdx i x)[j] = if h₁ : j < i then l[j] else if h₂ : j = i then x else l[j - 1]

theorem List.getElem?_insertIdx {α : Type u} {l : List α} {x : α} {i j : Nat} : (l.insertIdx i x)[j]? = if j < i then l[j]? else if j = i then if j ≤ l.length then some x else none else l[j - 1]?

theorem List.getElem?_insertIdx_of_lt {α : Type u} {l : List α} {x : α} {i j : Nat} (h : j < i) : (l.insertIdx i x)[j]? = l[j]?

theorem List.getElem?_insertIdx_self {α : Type u} {l : List α} {x : α} {i : Nat} : (l.insertIdx i x)[i]? = if i ≤ l.length then some x else none

theorem List.getElem?_insertIdx_of_gt {α : Type u} {l : List α} {x : α} {i j : Nat} (h : i < j) : (l.insertIdx i x)[j]? = l[j - 1]?

@[reducible, inline, deprecated List.getElem?_insertIdx_of_gt (since := "2025-02-04")]

abbrev List.getElem?_insertIdx_of_ge {α : Type u_1} {l : List α} {x : α} {i j : Nat} (h : i < j) : (l.insertIdx i x)[j]? = l[j - 1]?

Equations

• @List.getElem?_insertIdx_of_ge = @List.getElem?_insertIdx_of_gt